In this piece we will talk of small things and how they are. How small, you ask? Well, close to the scale of Planck’s h. This will mean the bits of stuff the world is made of: the bit of charge, the bit of light, and so on. These small bits are quite weird, not at all like the large stuff we are used to.

If you have a set of those bits, at each time it will be in a state. We write the state as a ket, as Paul taught us. The ket for state A looks like this: IA>. The ket has in it all one can know of the small bits we talk of. The state will change in time, of course. To say how much it will change per sec (its rate of change) we have a rule Er told us: i times the rate of change of the ket is H times the ket. H is a key thing; it keeps track of how much stuff is there is, the mass of each bit of stuff, which is the force that acts on each bit, and so on.

So we know how states change, fine. How do we get from this to a claim on what we will see in the lab? Here is where things are not like we are used to. In the large world we are used to, when we know the state of a thing we know that if we look at the thing we will see it in that state. But here, if we know the state we can’t tell in which state will we see it. We will have a chance to see it in a new state. If it was in IA>, it may be that we see IB> when we look. We can’t tell for sure. But we can tell what chance we have to see each state.

For that we use a rule that a guy called Max Born gave us. To get the chance to see state B when you look at some stuff that is in state A at that time, do some math called "ket A dot ket B", or IA> .IB>. What comes out of this math has to do with the chance to see B if the state is A. But it is not quite that, ‘cause the chance must be real and what we have now turns out to have i in it. Darned math! But just take the square of the real part and add to it the square of the part with i (that is, take the norm of what you have) and you will get the chance to see B if the state is A.

There are lots of things one can want to look at when one looks at stuff in the lab: Mass, charge, where bits are, how fast they move, and so on. For each of these things there are some states called “self states” of the thing. When the state is a self state of a thing, we can tell what we will see if we look at that thing. But in most states we can’t. As a case, if we add two self states IA> and IB> of a thing the new state IA> + IB> will not be a self state for that same thing. If this is the state, when we look at that thing we may see A or B, each with a chance of one half. Once we have looked, the state will be A or B; but not till we look.

This leads to the well known case of Er’s cat. If the state of some bits of stuff is IA> + IB>, and we put those bits in a box with a cat, and make things such that state A will in due time kill the cat, while state B lets it live, then the cat will not live nor die till we look in the box! Weird, huh?

I know you must want to say: “No way! For sure, the state was A or was B all the time, we just did not know which till we looked!” No such luck. If the state is IA> + IB>, then it may be that the A part and the B part “mix” and we can tell that the state was not “A or B” with a look at some things in which you see that mix. It is hard to do for a large thing, like a cat, ‘cause you would have to keep track of each bit of stuff; but it has been done for small things, and there is no clear line that breaks small from large.

Old Al did not like all this stuff. He was sure God did not play dice with the world. He and two pals came up with a thought to show all this must be wrong. Take two bits of stuff, say two light bits. Let the state be one in which both bits must have a thing not the same; say, one of them has “spin up” and one “spin down”, but the state makes not clear which has which spin. (Spin is like a turn ‘round that the bit may have; this turn may point up or down for each bit.) Let the two bits move a lot one each on its way, so they come to be far. Now look at the spin of the first bit. If you see it “up”, the spin of this bit has changed from an “up plus down” state to “up”, and the spin of bit two has changed at the same time from state “down plus up” to “down”. But how can the state of bit two change when we look at bit one, which is far from bit two? It makes no sense.

But it does. A smart guy called John Bell came up with a slick way to test this stuff, and it turned out that Old Al had been wrong for once. One is forced to grant that bits of stuff can “know” what goes on far, far from them, or else that in a sense things are not “out there” till we look at them! More and more weird, I say. It may be that some day we will make sense out of this. By now, guys are still not of one mind on what one should say. But, at the same time, with all this we can work out, know and grok lots of stuff. So it must be true. So when one starts to ask a lot, like “what does it mean”, some guys say: “Shut up and work!”

Coming next: Quantum Field Theory in words of one syllable. Quantum Gravity is much easier; one just needs to write: “What?”

13 Comments:

You sneaked in the fact that one can add states. ;) Most people don't pickup on this because adding is so elementary that we are well used to it, but in a sense (I've once read a beautiful quote to that effect which I can't find anymore) the whole weirdness of QM is contained in the statment "States are elements of a vector space" or, to keep with the constraint: "You can add two kets and get another ket".There is no apriori reason why that should be so! There is no physical meaning assosciated to the operation of adding two states,yet the result is supposed to be a physical state again!

You are absolutely right! In fact I have a couple of times summarized quantum mechanics in exactly that way ("States are elements of a vector space") for friends who studied things like economics or biology and thus know some linear algebra but little physics. Everything comes from that together with the Born rule for interpretation.

Have you seen the latest This Week's Finds? It has some cool stuff on toy versions/variations of quantum mechanics.

Thanks, old friend. (It's tough to say your name in just one sound!) I'm not sure if I will write more of this soon... it may be if I find a way to do Q. F. T. as I said, but it is more tough than just Q. M. I have not thought more than just how to say "Dick's graphs" so far...

Thank you both! And yes, Who is right; it would have been better to use "since" than "'cause", at least once and perhaps all the times, since "'cause" is a slight cheat. And you are right about the "norm" -in my defense I say that I find the "norm squared" appears more often than the "norm" in physics, and I would like to see it called "norm" instead of its square root.

Another mistake: " Let the two bits move a lot one each on its way" --> "each one on its way".

One part in which I cheated slightly is when describing the EPR-Bohm-Bell stuff -the most natural reading of what I wrote would be that the total state is "Iup down> + Idown up>", while in fact it is the singlet "Iup down> - Idown up>". I though explaining that properly would be too complicated.

As you might have guessed, I really like to do this stuff of explaining things in simple terms. So I would certainly love to write a textbook some day, or perhaps a popular science book. Your words give me encouragement in this dream...

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Thoughts on physics, maths, science, philosophy, and anything else that may cross my mind, plus news about my current life for distant friends, by an Argentinian in the second third year of his PhD at the University of Nottingham.

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